2.5: Impedance
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We need to remind ourselves of one other thing from electromagnetic theory before we can proceed, namely the meaning of impedance in the context of electromagnetic wave propagation. The impedance Z is merely the ratio E/H of the electric to the magnetic field. The SI units of E and H are V/m and A/m respectively, so the SI units of Z are V/A, or ohms, Ω. We are now going to see if we can express the impedance in terms of the permittivity and permeability of the medium in which an electromagnetic wave is travelling.
∇⋅D=ρ∇⋅B=0.∇×H=˙D+J.∇×E=−˙B.
In an isotropic, homogeneous, nonconducting, uncharged medium (such as glass, for example), the equations become:
∇⋅E=0∇⋅H=0∇×H=ϵ˙E.∇×H=−μ˙H.
If you eliminate H from these equations, you get
∇2E=ϵμ¨E,
which describes an electric wave of speed
1√ϵμ.
In free space, this becomes
1√ϵ0μ0
which is 2.998×108ms−1.
The ratio of the speeds in two media is
v1v2=n2n1=√ϵ2μ2ϵ1μ1,
and if, as is often the case, the two permeabilities are equal (to μ0), then
v1v2=n2n1=√ϵ2ϵ1.
In particular, if you compare one medium with a vacuum, you get: n=√ϵϵ0.
Light is a high-frequency electromagnetic wave. When a dielectric medium is subject to a high frequency field, the polarization (and hence D) cannot keep up with the electric field E. D lags behind E. This can be described mathematically by ascribing a complex value to the permittivity. The amount of lag depends, unsurprisingly, on the frequency - i.e. on the color - and so the permittivity and hence the refractive index depends on the wavelength of the light. This is dispersion.
If instead you eliminate E from Maxwell’s equations, you get
∇2H=ϵμ¨H.
This is a magnetic wave of the same speed.
If you eliminate the time between ??? and ???, you find that EH=√μϵ, which, in free space, has the value √μ0ϵ0=377Ω, which is the impedance of free space. In an appropriate context I may use the symbol Z0 to denote the impedance of free space, and the symbol Z to denote the impedance of some other medium.
The ratio of the impedances in two media is
Z1Z2=√ϵ2μ1ϵ1μ2,
and if, as is often the case, the two permeabilities are equal (to μ0), then
Z1Z2=√ϵ2ϵ1=n2n1=v1v2.
We shall be using this result in what follows.